Exceptional Polynomials Research Articles

Recurrence Relations of Exceptional Laurent Biorthogonal Polynomials

Recurrence Relations of Exceptional Laurent Biorthogonal Polynomials

Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.

Matrix exceptional Laguerre polynomials

AbstractWe give an analog of exceptional polynomials in the matrix‐valued setting by considering suitable factorizations of a given second‐order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix‐valued exceptional Laguerre polynomials of arbitrary size.

Time and band limiting for exceptional polynomials

The “time-and-band limiting” commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones.Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the “bispectral property”. As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.

The Black–Scholes equation in finance: Quantum mechanical approaches

In this paper, the Black–Scholes equation of the option pricing theory in order to minimize the risk through the stocks is studied. The solutions are obtained in terms of exceptional Laguerre polynomials. Moreover, higher-order supesymmetric representations are studied with a special case of third order. The Darboux transformation of the heat equation linked to the Black–Scholes system is given and a new potential is shown.

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

AbstractA formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second‐order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional‐type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state‐deletion and state‐addition occur.

The single-indexed exceptional Krawtchouk polynomials

ABSTRACT The Darboux transformations of Krawtchouk polynomials are investigated and all possible exceptional Krawtchouk polynomials obtainable from a single-step Darboux transformation are considered. The properties of these exceptional Krawtchouk polynomials including the Diophantine ones and the recurrence relations are obtained.

Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials

Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials, and their properties are explored. In particular, fractional revival is shown to take place in the corresponding quantum walks.

Supersymmetry and shape invariance of exceptional orthogonal polynomials

We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger equation to define “Hamiltonians” enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the solubility of the differential equations associated with these polynomials.

R-fat linearized polynomials over finite fields

r-fat polynomials are a natural generalization of scattered polynomials. They define linear sets of the projective line PG(1,qn) of rank n with r points of weight larger than one. Using techniques on algebraic curves and function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r>0. We completely determine the possible values of r when considering linearized polynomials over Fq4 and we also provide one family of 1-fat polynomials in PG(1,q5). Furthermore, we investigate LP-polynomials (i.e. polynomials of type f(x)=x+δxq2s∈Fqn[x], gcd⁡(n,s)=1), determining the spectrum of values r for which such polynomials are r-fat.

Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials

We extend and generalize the construction of Sturm–Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the ‘−2x/3’ hierarchy of solutions to the fourth Painlevé transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes constructed in terms of the generalized Okamoto polynomials. The third-order ladder operators of the system reveal that the complete set of eigenfunctions is decomposed as a union of three disjoint sequences of solutions, generated from a set of three-term recurrence relations. We also identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite polynomial.

Spin and pseudospin symmetries in radial Dirac equation and exceptional hermite polynomials

We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

AbstractWe construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second‐order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum‐mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

Statistical mechanics of DNA mutation using SUSY quantum mechanics

This paper investigates the deoxyribonucleic acid (DNA) denaturation through statistical mechanics and demonstrates that the exceptional polynomials lead to DNA mutation. A DNA model with two chains connected by the Morse potential representing the H bonds is considered and the partition function for this model is computed. The partition function is converted into a Schrödinger-like equation. The techniques of SUSY quantum mechanics are used to model the DNA mutation. The thermal denaturation of DNA for each mutated state is also computed.

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.

Numerical study of the SWKB condition of novel classes of exactly solvable systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials

Exceptional Laguerre and Jacobi polynomials p n ( x ) p_n(x) are bispectral, in the sense that as functions of the continuous variable x x , they are eigenfunctions of a second order differential operator and as functions of the discrete variable n n , they are eigenfunctions of a higher order difference operator (the one defined by any of the recurrence relations they satisfy). In this paper, under mild conditions on the sets of parameters, we characterize the algebra of difference operators associated to the higher order recurrence relations satisfied by the exceptional Laguerre and Jacobi polynomials.

General solution of the exceptional Hermite differential equation and its minimal surface representation

The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition $\lambda = (1)$ and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the general analytic solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is complemented by non-polynomial solutions. We present an implementation of the obtained results for the surfaces expressed in terms of the general solution making use of the classical Enneper-Weierstrass formula for the immersion in the Euclidean space $\mathbb{E}^3$, leading to minimal surfaces. Three-dimensional displays of these surfaces are presented.

Corrigendum on the proof of completeness for exceptional Hermite polynomials

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm–Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Grünbaum and Oblomkov.

Coherent states for rational extensions and ladder operators related to infinite-dimensional representations

The systems we consider are rational extensions of the harmonic oscillator, the truncated oscillator and the radial oscillator. The wavefunctions for the extended states involve exceptional Hermite polynomials for the oscillator and truncated oscillator and exceptional Laguerre polynomials for the radial oscillator. In all cases it is possible to construct ladder operators that have infinite-dimensional representations of their polynomial Heisenberg algebras and couple all levels of the systems. We construct Barut-Girardello coherent states in all cases, eigenvectors of the respective annihilation operators with complex eigenvalues. Then we calculate their physical properties to look for classical or non-classical behaviour.